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1 VECTOR SPACES 359

(1.4) for each x e E there exists an element — x e E such that x 4- (— x) = 0
(II. 1) (A + /*)x = Ax + //x

(II.4) 1 - jc = x

(in these conditions, x, y, z are arbitrary elements of E and A, JJL are arbitrary
elements of K).

Conditions (LI) to (1.4) express that E is a commutative group with respect
to addition. It follows that, if (Xi)_leH is any finite family of elements of E,
the sum ]Tx_f is unambiguously defined. (We recall the convention that

ieH

From (1.3) and (IL2) we deduce that Ax = A(x + 0) = Ax + AO, and there-
fore AO = 0 for all X e K. From (II. 1) it follows that Ax = (A + 0)x = Ax + Ox,
so that Ox = 0 for all x e E. Finally, from these two relations and (L4), (II. 1),
and (II.2) we deduce that Ax + A(-x) - 10 = 0, and Ax + (-A)x = Ox = 0,
so that (-A)x = A(-x) = -(Ax).

The elements of E are often called vectors.

(A.1.2) An additive group consisting of the element 0 alone is a vector
space: the scalar multiplication is the unique mapping of K x {0} into {0}.
The field K is a vector space over itself, the scalar multiplication being
multiplication in K. If K' is a subfield of K and if E is a vector space over K,
then E is also a vector space over K' if we define the scalar multiplication
to be the restriction to K' x E of the mapping (A, x) -* Ax.

(A.1.3) If E is a vector space, a vector subspace (or simply subspace) of E
is defined to be any subset F of E such that the relations x e F, y e F imply
Ax + w e F, for all scalars A, \JL.

The restriction to F x F (resp. K x F) of the addition (resp. scalar multi-
plication) in E is a mapping into F, and therefore F is a vector space with
respect to these mappings. This justifies the terminology.

If (x_t.)i ^ i^_n is ^anY fin**⁶ family of vectors in F, and if (A_t\ ^ ^_n is a family

of scalars, then by induction on n we see that J] A_f x_f e F. A vector of the

i-l
n

form £ A_fx_£ (where x_£ e E and A_te K) is called a linear combination of the

1 = 1
x_f . (A linear combination is always a finite sum, even if E is, for example, a

normed space (Section 5.1) in which certain infinite sums are defined (Section
5.3).)



	1 VECTOR SPACES 359

	(1.4) for each x e E there exists an element — x e E such that x 4- (— x) = 0 (II. 1) (A + /*)x = Ax + //x

	(II.4) 1 - jc = x (in these conditions, x, y, z are arbitrary elements of E and A, JJL are arbitrary elements of K). Conditions (LI) to (1.4) express that E is a commutative group with respect to addition. It follows that, if (Xi)_leH is any finite family of elements of E, the sum ]Tx_f is unambiguously defined. (We recall the convention that

	ieH

	From (1.3) and (IL2) we deduce that Ax = A(x + 0) = Ax + AO, and there- fore AO = 0 for all X e K. From (II. 1) it follows that Ax = (A + 0)x = Ax + Ox, so that Ox = 0 for all x e E. Finally, from these two relations and (L4), (II. 1), and (II.2) we deduce that Ax + A(-x) - 10 = 0, and Ax + (-A)x = Ox = 0, so that (-A)x = A(-x) = -(Ax). The elements of E are often called vectors.

	(A.1.2) An additive group consisting of the element 0 alone is a vector space: the scalar multiplication is the unique mapping of K x {0} into {0}. The field K is a vector space over itself, the scalar multiplication being multiplication in K. If K' is a subfield of K and if E is a vector space over K, then E is also a vector space over K' if we define the scalar multiplication to be the restriction to K' x E of the mapping (A, x) -* Ax.

	(A.1.3) If E is a vector space, a vector subspace (or simply subspace) of E is defined to be any subset F of E such that the relations x e F, y e F imply Ax + w e F, for all scalars A, \JL. The restriction to F x F (resp. K x F) of the addition (resp. scalar multi- plication) in E is a mapping into F, and therefore F is a vector space with respect to these mappings. This justifies the terminology. If (x_t.)i ^ i^_n is ^anY fin*⁶ family of vectors in F, and if (A_t\ ^ ^_n* is a family

	of scalars, then by induction on n we see that J] A_f x_f e F. A vector of the i-l n form £ A_fx_£ (where x_£ e E and A_te K) is called a linear combination of the 1 = 1 x_f . (A linear combination is always a finite sum, even if E is, for example, a normed space (Section 5.1) in which certain infinite sums are defined (Section 5.3).)